response_original:
<think>
Let's analyze the problem step by step:
- Total dishes: 36
- Dishes with mango salsa: 3
- Dishes with fresh mangoes: "a sixth" of 36 = 36 / 6 = 6
- Dishes with mango jelly: 1
First, we need to be careful: these categories may have overlap. For example, a dish could have both mango salsa and fresh mango, but the problem doesn't state whether the dishes overlap or not.
However, since the problem gives:
- Mango salsa is on three dishes (doesn't specify "in addition to" fresh mangoes), but also says the restaurant "features mango salsa on three of its dishes, fresh mangoes in a sixth..." and "mango jelly in one".
When presented in this way in counting problems, unless specified, we'll assume 3 dishes with mango salsa, 6 dishes with fresh mango (some may already be counted in salsa group), and 1 with mango jelly; but they might overlap.
But, since only two of the dishes with fresh mango can be picked out and thus he can eat those, the rest with mango he cannot eat. So, he can eat 2 of the fresh mango dishes (by picking out mango), and must avoid all other mango dishes.
So, calculate the number of "unavailable" dishes (with mango) and subtract.
Number of dishes with mango in any form:
- With fresh mango: 6 (Oliver can eat 2 if he picks out mango, so only 4 strictly off-limits)
- With mango salsa: 3 (no indication he can eat these, so all off-limits)
- With mango jelly: 1 (off-limits)
But these may have overlap (i.e., a dish with both salsa and jelly, etc).
Let's suppose:
Let A = Dishes with mango salsa = 3
Let B = Dishes with fresh mango = 6
Let C = Dishes with mango jelly = 1
We want the number of unique dishes that have any mango, i.e., |A ∪ B ∪ C|
But information about overlaps is missing. Since nothing indicates overlaps, and since these are different mango preparations, we'll assume no overlap (i.e., worst case for Oliver).
Sum:
- Mango salsa only: 3
- Fresh mango only: 6
- Mango jelly only: 1
Total mango dishes: 3 + 6 + 1 = 10
BUT, if there are overlaps (e.g., a dish with both mango salsa and jelly), that would reduce the count of unique mango-dishes, increasing the number safe for Oliver. But since the problem does not specify, the standard is to assume "worst case" for Oliver: assume no overlap.
However, he can eat 2 of the 6 dishes with fresh mango because he can pick it out.
So, of the (mango) dishes, he can eat 2 from the fresh mango group. The rest of the mango dishes are unavailable.
So, the number of dishes he cannot eat:
- Mango salsa: 3 (off-limits)
- Fresh mango: 4 (cannot eat, as out of 6 with fresh mangoes, he can pick out mango in only 2, so 4 are off-limits)
- Mango jelly: 1 (off-limits)
Wait: are any dishes with mango salsa or mango jelly also included in "dishes with fresh mango"? We have to think carefully.
If there is overlap, that would mean the "unavailable" count is less (since some dishes are double or triple counted). If we sum the "unavailable dishes" as 3+4+1=8.
Another way: Of the total 36 dishes:
- Dishes with fresh mango = 6, but Oliver can eat 2, so 4 are off-limits: 4
- Dishes with mango salsa: 3 (do these overlap with the fresh mango dishes? Possibly. But unless specified, normally assumed as separate items.)
- Dishes with mango jelly: 1 (do these overlap with either of the above? Not specified.)
But, the problem's standard approach is to assume no overlap unless specified.
Thus, unavailable dishes:
4 (fresh mango) + 3 (mango salsa) + 1 (mango jelly) = 8
Thus, dishes available for Oliver:
36 - 8 = 28
</think>
<answer>28</answer>